The sequence $x_1, x_2, x_3, ...$ is such that $x_n = 1/n - 1/(n+1)$ . What is the sum of the first 100 terms of the sequence?

Correct answer marked below

201/100

99/100

100/101

Correct

1/10000

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Comments

HiyoriAug 20

This is a classic telescoping sum. I just wrote out the first few terms and saw the cancellation.

KatieAug 23

Exactly! Once you see that pattern, it's smooth sailing.

LostMay 19

Wait, how do you know it telescopes? I tried plugging in n=1 and got 1/2, but then I got stuck.

TrangMay 24

Look at x1 + x2: (1/1 - 1/2) + (1/2 - 1/3) = 1 - 1/3. The middle terms cancel.

LubainaMay 29

Thanks! That makes sense now.

ZaberMay 17

I got 100/101. Did anyone else?

MahakJun 18

For me, the hardest part was remembering that the sum of the first n terms is 1 - 1/(n+1). Easy to mix up the indices.

BerkayJun 29

Same here. I almost picked 99/100 because I used n=99 instead of n=100.

What this question tests

Your ability to spot and extend arithmetic and geometric sequences and work with recursive definitions.

Off-by-one errors in indexing (a₀ vs a₁) or confusing an arithmetic difference with a geometric ratio.

The sequence x1,x2,x3,...x_1, x_2, x_3, ...x1​,x2​,x3​,... is such that xn=1/n−1/(n+1)x_n = 1/n - 1/(n+1)xn​=1/n−1/(n+1). What is the sum of the first 100 terms of the sequence?